defectDist_self
plain-language theorem explainer
The theorem establishes that defect distance vanishes when its two arguments coincide for any positive real. Researchers building cost algebras or defect metrics in Recognition Science cite it as the identity axiom. The term-mode proof reduces the claim by definition substitution to the normalization J(1) = 0.
Claim. For every positive real number $x > 0$, the defect distance satisfies $d(x,x) = 0$, where $d(x,y) := J(x/y)$ and $J$ is the cost function normalized by $J(1) = 0$.
background
Defect distance is introduced in CostAlgebra as the function $d(x,y) = J(x/y)$ for positive reals $x,y$. It quantifies the cost of deviation between two values under the J-cost induced by the Recognition Composition Law. The upstream result J_at_one states that the multiplicative identity carries zero cost, i.e., $J(1) = 0$, which follows from the unit normalization in the functional equation module.
proof idea
The proof unfolds the definition of defectDist to reach $J(x/x)$. The hypothesis $x > 0$ permits rewriting the ratio as 1 by the div_self lemma. The claim is then discharged directly by the theorem J_at_one.
why it matters
This identity anchors the defect distance as the zero element of the cost algebra and supports the symmetry and non-negativity properties required for the Recognition Composition Law. It supplies the base case for constructions that later connect to the phi-ladder mass formulas and the alpha band bounds. The result closes a foundational step before extensions to empirical programs or ledger structures.
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